Reliable measurements of the three-dimensional radial distribution function for cloud droplets are desired to help characterize microphysical processes that depend on local drop environment. Existing numerical techniques to estimate this three-dimensional radial distribution function are not well suited to in situ or laboratory data gathered from a finite experimental domain. This paper introduces and tests a new method designed to reliably estimate the three-dimensional radial distribution function in contexts in which (i) physical considerations prohibit the use of periodic boundary conditions and (ii) particle positions are measured inside a convex volume that may have a large aspect ratio. The method is then utilized to measure the three-dimensional radial distribution function from laboratory data taken in a cloud chamber from the Holographic Detector for Clouds (HOLODEC).

Cloud droplet clustering is relevant to physical processes like
condensational growth (e.g.,

Most of the in situ studies cited above have utilized airplane-mounted cloud
probes that report cloud particle positions in a long, thin, pencil-beam-like
volume. For example, the sample volume of the forward scattering spectrometer
probe has a cross section of about 0.13 mm

The most direct and assumption-free way to detect cloud particle clustering
is with an instrument that is capable of recording precise particle locations
in all three spatial dimensions. This can be carried out with a holographic image of
a cloud volume. Some previous holographic studies that explicitly examined
three-dimensional cloud particle spatial distributions have been published
(see, e.g.,

Fortunately, both computational and measurement hardware capabilities, as
well as analysis methods, have improved immensely over the last decade,
finally bringing holography to a fully digital state that allows for data
collection and processing over entire field projects (e.g.,

There are many different mathematical tools utilized to characterize the
droplet clustering among cloud droplets, each with their own strengths and
weaknesses (see, e.g.,

Although calculation of the three-dimensional radial distribution function
from experimentally measured particle position data should be possible,
properly accounting for the effects of the edges of the measurement volume
can be tricky

The remainder of this paper (i) reintroduces the radial distribution function, (ii) presents the methods typically used to estimate the radial distribution function in different experimental and numerical contexts, (iii) outlines the challenges in utilizing these existing methods for experimental data from modern digital holographic images, (iv) presents and tests a new numerical method to calculate the radial distribution function under realistic experimental conditions, and (v) applies this method to real data taken by a digital holographic instrument in a cloud chamber.

The radial distribution function is one of the most widely used approaches
for characterizing particle clustering in turbulent flows

Here, we draw on the introduction given in

In contexts in which the spatial coordinates of each member of a population
of particles are resolved, the radial distribution function at scale

Calculation of

The above formula has been used in most previous experimental studies
computing the radial distribution functions for cloud droplets. In principle,
this result can then be used to estimate the three-dimensional radial
distribution function following the method outlined in

The three-dimensional radial distribution function

When searching for another cloud droplet separated by scale

A cartoon of this process (shown in two dimensions) can be viewed in Fig.

A two-dimensional cartoon of the different ways of traditionally
dealing with domain edges when computing the radial distribution function.
Panel

Unfortunately, the technique described in Fig.

The simplest possible solution, albeit the most drastic, in trying to
estimate the radial distribution function for finite experimental volumes is
to ignore these edge effects entirely. For the cartoon in Fig.

Much like in the one-dimensional case, the effects of the edges sometimes can
be small enough to make this a minor concern. When the scale of interest

As noted earlier, this is a problem that has received attention for at least
35 years

Note that

Figure

Another cartoon of the guard area technique used to
estimate the radial distribution function. Note that the fraction of the
particles contributing to the sum in Eq. (

The guard area approach

Here, we introduce an alternative edge-correction strategy inspired by

This is very similar to Eq. (

The challenging part of the method is to find

There are multiple ways to generate the proposed look-up table. In this
study, we have populated the interior of the measurement volume with a
regular dense grid with grid spacing

Conceptually, the algorithm uses the

The effective volume method allows for any investigator-chosen values of

The effective volume method described above was implemented for two different geometries – a cubical geometry (to allow for useful comparisons to the well-known and frequently utilized guard area technique) as well as for an applied geometry to match a real instrument. For each geometry, we present two tests: a homogeneous Poisson distribution and a Matérn cluster process.

A homogeneous Poisson distribution is the gold standard of spatial
randomness. Within a homogeneous Poisson distribution, all particles are
placed independently with a spatial density function uniform over the
measurement domain. By construction,

A Matèrn cluster process (see, e.g.,

Both distributions described above were simulated within a unit cube with both guard area and effective volume computation methods.

For the guard area computation method, a fixed

The effective volume computation method was computed by creating a look-up table for a cubical volume. Due to the symmetry of the volume, only one octant of the cube had to be included in the look-up table. To minimize the size of the look-up table required, the density of the tessellation of the cubical measurement volume was varied depending on the distance to the boundary – with points near the boundary having the densest collection of look-up table entries.

A total of 100 simulations of each volume were averaged together and the
results are displayed in Fig.

In this case, the guard area approach involves summing over less than
half (on average 48.8 %) of particles in the measurement volume, which
can help to explain the larger scatter of observed

First
verification of the method for calculating

Although the effective volume approach introduced here performs approximately
as well as the more traditional guard area approach in cubical volumes, the
development of the new method was primarily motivated by a desire to estimate
the radial distribution function in contexts in which the guard area approach
will not work. As noted previously, when estimates of

An example of an instrument that is subject to these limitations and is relevant for studying cloud particle clustering is the Holographic Detector for Clouds (HOLODEC).

HOLODEC is an in-line digital holography instrument explicitly designed to
explore cloud microstructure

A processed HOLODEC hologram reports droplet positions in a volume that is
approximately 1 cm

A two-dimensional cartoon of the HOLODEC sample volume is shown in
Fig.

A two-dimensional cartoon of the HOLODEC sample volume (not to scale). The leftmost vertical line in the figure indicates the hologram plane. The light grey region indicates areas of maximum sensor sensitivity (the slope of the angled lines marking the edge of the light grey region has been greatly magnified for aid in visualization). The vertical lines 14 and 158 mm from the left edge of the figure mark the positions of the optical windows; near these windows there is evidence of artificially generated particles due to instrument-induced particle fragmentation. The volume simulated here corresponds to the darker central grey rectangle (parallelepiped in 3-D), where the instrument retains approximately uniform sensitivity, particle locations and sizes are believed to be accurate, and the number of small particles generated due to fragmentation on the instrument is believed to be negligible.

The same two tests (homogeneous Poisson distribution and Matérn cluster
process) were simulated within the parallelepiped sample volume of the
HOLODEC. A new look-up table for this geometry was generated and used to
calculate

Verification that the effective volume
method for calculating

In general, the agreement between the theoretical expressions for

To test the claim made above, a proof-of-principle analysis was completed
using real HOLODEC data acquired inside a laboratory cloud chamber driven by
Rayleigh–Bénard convection

The radial distribution functions of the eight holograms with the largest numbers
of detected drops are shown in Fig.

The measured radial distribution functions for eight different holograms and their mean for HOLODEC data taken in the cloud chamber. Clearly sampling variability is still pronounced at small spatial scales, but some evidence of scale-dependent clustering seems possible.

These single-hologram results are noisy due to the sampling uncertainty
(especially for the smallest spatial scales), but it is clear that there is
some evidence of scale-dependent clustering revealed for

Actually measuring the three-dimensional radial distribution function for in situ
flight data will still be challenging, even with the aid of the algorithm
introduced here. The radial distribution function curves shown in Fig.

Understanding the effects of cloud particle clustering on microphysical processes requires reliable estimation of the three-dimensional radial distribution function. Previous studies have obtained this information by utilizing one-dimensional measurements of cloud particle positions to infer scale-dependent clustering, but these methods have been shown to carry large uncertainties. In the hope of finding an alternative way of characterizing cloud particle clustering without such restrictive underlying assumptions and/or uncertainties, measurement of the radial distribution function for in situ data in three dimensions is desired.

Comparing measurements with theory and numerical simulation relies on
estimating

Here, a new method was introduced that explicitly considers each particle's position within the measurement volume in the radial distribution function computation. This method allows for calculating the radial distribution function for scales larger than the shortest physical dimension of the measurement volume and makes more optimal use of the measured data. This effective volume method was tested in two different geometries, compared to standard computational methods with simulated data in a unit cube, and validated in a more realistic sampling scenario.

Preliminary results confirm that use of the effective volume method should enable the use of airborne digital holography data to compute in situ three-dimensional radial distribution functions for cloud droplets.

The HOLODEC data associated with the analysis in
Sect.

The effective volume method to calculate the radial distribution function relies on two codes – one to generate a look-up table for the measurement volume, and another to use the look-up table and data to compute the radial distribution function. This appendix outlines the basic structure utilized for each of these codes.

Required inputs from the user include the following: physical domain of sample volume, set of radii

Tessellate the interior of the sample volume domain at scale

For each radius

Tessellate an

For each radius

Compute the factor

Required inputs include the same set of inputs utilized to generate the
look-up table and

Load the look-up table.

For each radius

For each particle

count the number of other particles that are between

identify the closest entry in the look-up table

assign d

use d

Compute and return

The authors declare that there is no conflict of interest.

This work was supported by the US National Science Foundation through grants
AGS-1532977 (MLL) and AGS-1623429 (RAS). Special thanks to Alexander
Kostinski, Susanne Glienke, and Neel Desai for helpful discussions and help
with accessing and interpreting the HOLODEC data from the